Properties

Label 810.4.e.u
Level $810$
Weight $4$
Character orbit 810.e
Analytic conductor $47.792$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 810.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(47.7915471046\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 90)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{6} + 2) q^{2} - 4 \zeta_{6} q^{4} + 5 \zeta_{6} q^{5} + (14 \zeta_{6} - 14) q^{7} - 8 q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \zeta_{6} + 2) q^{2} - 4 \zeta_{6} q^{4} + 5 \zeta_{6} q^{5} + (14 \zeta_{6} - 14) q^{7} - 8 q^{8} + 10 q^{10} + (6 \zeta_{6} - 6) q^{11} - 68 \zeta_{6} q^{13} + 28 \zeta_{6} q^{14} + (16 \zeta_{6} - 16) q^{16} + 78 q^{17} + 44 q^{19} + ( - 20 \zeta_{6} + 20) q^{20} + 12 \zeta_{6} q^{22} - 120 \zeta_{6} q^{23} + (25 \zeta_{6} - 25) q^{25} - 136 q^{26} + 56 q^{28} + (126 \zeta_{6} - 126) q^{29} + 244 \zeta_{6} q^{31} + 32 \zeta_{6} q^{32} + ( - 156 \zeta_{6} + 156) q^{34} - 70 q^{35} - 304 q^{37} + ( - 88 \zeta_{6} + 88) q^{38} - 40 \zeta_{6} q^{40} + 480 \zeta_{6} q^{41} + (104 \zeta_{6} - 104) q^{43} + 24 q^{44} - 240 q^{46} + (600 \zeta_{6} - 600) q^{47} + 147 \zeta_{6} q^{49} + 50 \zeta_{6} q^{50} + (272 \zeta_{6} - 272) q^{52} - 258 q^{53} - 30 q^{55} + ( - 112 \zeta_{6} + 112) q^{56} + 252 \zeta_{6} q^{58} - 534 \zeta_{6} q^{59} + (362 \zeta_{6} - 362) q^{61} + 488 q^{62} + 64 q^{64} + ( - 340 \zeta_{6} + 340) q^{65} + 268 \zeta_{6} q^{67} - 312 \zeta_{6} q^{68} + (140 \zeta_{6} - 140) q^{70} - 972 q^{71} + 470 q^{73} + (608 \zeta_{6} - 608) q^{74} - 176 \zeta_{6} q^{76} - 84 \zeta_{6} q^{77} + (1244 \zeta_{6} - 1244) q^{79} - 80 q^{80} + 960 q^{82} + (396 \zeta_{6} - 396) q^{83} + 390 \zeta_{6} q^{85} + 208 \zeta_{6} q^{86} + ( - 48 \zeta_{6} + 48) q^{88} - 972 q^{89} + 952 q^{91} + (480 \zeta_{6} - 480) q^{92} + 1200 \zeta_{6} q^{94} + 220 \zeta_{6} q^{95} + ( - 46 \zeta_{6} + 46) q^{97} + 294 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 4 q^{4} + 5 q^{5} - 14 q^{7} - 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 4 q^{4} + 5 q^{5} - 14 q^{7} - 16 q^{8} + 20 q^{10} - 6 q^{11} - 68 q^{13} + 28 q^{14} - 16 q^{16} + 156 q^{17} + 88 q^{19} + 20 q^{20} + 12 q^{22} - 120 q^{23} - 25 q^{25} - 272 q^{26} + 112 q^{28} - 126 q^{29} + 244 q^{31} + 32 q^{32} + 156 q^{34} - 140 q^{35} - 608 q^{37} + 88 q^{38} - 40 q^{40} + 480 q^{41} - 104 q^{43} + 48 q^{44} - 480 q^{46} - 600 q^{47} + 147 q^{49} + 50 q^{50} - 272 q^{52} - 516 q^{53} - 60 q^{55} + 112 q^{56} + 252 q^{58} - 534 q^{59} - 362 q^{61} + 976 q^{62} + 128 q^{64} + 340 q^{65} + 268 q^{67} - 312 q^{68} - 140 q^{70} - 1944 q^{71} + 940 q^{73} - 608 q^{74} - 176 q^{76} - 84 q^{77} - 1244 q^{79} - 160 q^{80} + 1920 q^{82} - 396 q^{83} + 390 q^{85} + 208 q^{86} + 48 q^{88} - 1944 q^{89} + 1904 q^{91} - 480 q^{92} + 1200 q^{94} + 220 q^{95} + 46 q^{97} + 588 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/810\mathbb{Z}\right)^\times\).

\(n\) \(487\) \(731\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
271.1
0.500000 0.866025i
0.500000 + 0.866025i
1.00000 + 1.73205i 0 −2.00000 + 3.46410i 2.50000 4.33013i 0 −7.00000 12.1244i −8.00000 0 10.0000
541.1 1.00000 1.73205i 0 −2.00000 3.46410i 2.50000 + 4.33013i 0 −7.00000 + 12.1244i −8.00000 0 10.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 810.4.e.u 2
3.b odd 2 1 810.4.e.a 2
9.c even 3 1 90.4.a.b 1
9.c even 3 1 inner 810.4.e.u 2
9.d odd 6 1 90.4.a.e yes 1
9.d odd 6 1 810.4.e.a 2
36.f odd 6 1 720.4.a.e 1
36.h even 6 1 720.4.a.t 1
45.h odd 6 1 450.4.a.c 1
45.j even 6 1 450.4.a.m 1
45.k odd 12 2 450.4.c.g 2
45.l even 12 2 450.4.c.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.4.a.b 1 9.c even 3 1
90.4.a.e yes 1 9.d odd 6 1
450.4.a.c 1 45.h odd 6 1
450.4.a.m 1 45.j even 6 1
450.4.c.f 2 45.l even 12 2
450.4.c.g 2 45.k odd 12 2
720.4.a.e 1 36.f odd 6 1
720.4.a.t 1 36.h even 6 1
810.4.e.a 2 3.b odd 2 1
810.4.e.a 2 9.d odd 6 1
810.4.e.u 2 1.a even 1 1 trivial
810.4.e.u 2 9.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(810, [\chi])\):

\( T_{7}^{2} + 14T_{7} + 196 \) Copy content Toggle raw display
\( T_{11}^{2} + 6T_{11} + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$7$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$11$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$13$ \( T^{2} + 68T + 4624 \) Copy content Toggle raw display
$17$ \( (T - 78)^{2} \) Copy content Toggle raw display
$19$ \( (T - 44)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 120T + 14400 \) Copy content Toggle raw display
$29$ \( T^{2} + 126T + 15876 \) Copy content Toggle raw display
$31$ \( T^{2} - 244T + 59536 \) Copy content Toggle raw display
$37$ \( (T + 304)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 480T + 230400 \) Copy content Toggle raw display
$43$ \( T^{2} + 104T + 10816 \) Copy content Toggle raw display
$47$ \( T^{2} + 600T + 360000 \) Copy content Toggle raw display
$53$ \( (T + 258)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 534T + 285156 \) Copy content Toggle raw display
$61$ \( T^{2} + 362T + 131044 \) Copy content Toggle raw display
$67$ \( T^{2} - 268T + 71824 \) Copy content Toggle raw display
$71$ \( (T + 972)^{2} \) Copy content Toggle raw display
$73$ \( (T - 470)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 1244 T + 1547536 \) Copy content Toggle raw display
$83$ \( T^{2} + 396T + 156816 \) Copy content Toggle raw display
$89$ \( (T + 972)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 46T + 2116 \) Copy content Toggle raw display
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